The \(L^p - L^q\)  boundedness for \(T_\alpha \)  with \(\alpha \in (0,\frac{n+1}{2}]\)  was studied by several authors ([1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}). We will make use of it to prove the Carleman inequality (REF ) for all admissible \(p,q\)  in the following section. Here, let us for the moment digress and introduce some convenient notations to describe the set of \(p,q\)  for which \(\Vert T_\alpha \Vert _{p\rightarrow q}\)  is bounded.
